Question: $\sum\limits_{n=2}^{\infty}\dfrac{\ln(n)}{n^2}$ Does the integral test apply to the series? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Explanation: $\dfrac{\ln(x)}{x^2}$ is continuous and positive for all $x\geq 2$. To find whether it's always decreasing for $x\geq2$, let's consider its derivative. $\dfrac{d}{dx}\left(\dfrac{\ln(x)}{x^2}\right)=\dfrac{1-2\ln(x)}{x^3}$ For $x\geq 2>\sqrt e$, we have $2\ln(x)>1$, which means $1-2\ln(x)$ is negative. So $\dfrac{d}{dx}\left(\dfrac{\ln(x)}{x^2}\right)$ is negative for all $x\geq 2$, which means $\dfrac{\ln(x)}{x^2}$ is decreasing. In conclusion, the integral test does apply to the series.